Virtually Abelian Subgroups of IA n ( Z / 3 ) Are Abelian

Abstract

When studying subgroups of $\mathsf{Out}\left(F_n\right)$, one often replaces a given subgroup $\mathcal{H}$ with one of its finite index subgroups ${\mathcal{H}}_0$ so that virtual properties of $\mathcal{H}$ become actual properties of ${\mathcal{H}}_0$. In many cases, the finite index subgroup is ${\mathcal{H}}_0=\mathcal{H}\cap {IA}_n(\mathbb{Z}/3)$. For which properties is this a good choice? Our main theorem states that being abelian is such a property. Namely, every virtually abelian subgroup of ${IA}_n(\mathbb{Z}/3)$ is abelian.